using&randomiza-on&methods&to& … · 2014-10-29 ·...

Using Randomiza-on Methods to Build Conceptual Understanding in

Sta-s-cal Inference: Day 1

Lock, Lock, Lock, Lock, and Lock MAA Minicourse – Joint Mathema6cs Mee6ngs

San Diego, CA January 2013

The Lock5 Team

Robin St. Lawrence

Dennis Iowa State

Eric Duke

Kari Duke

PaD St. Lawrence

Introduc6ons:

Name Ins6tu6on

Schedule: Day 1 Wednesday, 1/9, 9:00 – 11:00 am

1. Introduc-ons and Overview

2. Bootstrap Confidence Intervals •  What is a bootstrap distribu6on? •  How do we use bootstrap distribu6ons to build understanding of confidence intervals? •  How do we assess student understanding when using this approach?

3. GeDng Started on Randomiza-on Tests •  What is a randomiza6on distribu6on? •  How do we use randomiza6on distribu6ons to build understanding of p-‐values?

4. Minute Papers

Schedule: Day 2 Friday, 1/11, 9:00 – 11:00 am

5. More on Randomiza-on Tests •  How do we generate randomiza6on distribu6ons for various sta6s6cal tests?

•  How do we assess student understanding when using this approach?

6. Connec-ng Intervals and Tests 7. Connec-ng Simula-on Methods to Tradi-onal

8. Technology Op-ons •  Brief soSware demonstra6on (Minitab, Fathom, R, Excel, ...) – pick one!

9. Wrap-‐up •  How has this worked in the classroom? •  Par6cipant comments and ques6ons

10. Evalua-ons

Why use Randomiza6on Methods?

These methods are great for teaching sta6s6cs…

(the methods 6e directly to the key ideas of sta6s6cal inference

so help build conceptual understanding)

And these methods are becoming increasingly important for doing

sta6s6cs.

It is the way of the past…

"Actually, the statistician does not carry out this very simple and very tedious process [the randomization test], but his conclusions have no justi;ication beyond the fact that they agree with those which could have been arrived at by this elementary method."

-‐-‐ Sir R. A. Fisher, 1936

… and the way of the future “... the consensus curriculum is still an unwitting prisoner of history. What we teach is largely the technical machinery of numerical approximations based on the normal distribution and its many subsidiary cogs. This machinery was once necessary, because the conceptually simpler alternative based on permutations was computationally beyond our reach. Before computers statisticians had no choice. These days we have no excuse. Randomization-‐based inference makes a direct connection between data production and the logic of inference that deserves to be at the core of every introductory course.”

-‐-‐ Professor George Cobb, 2007

(see full TISE article by Cobb in your binder)

Ques6on

Do you teach Intro Stat? A. Very regularly (most semesters) B. Regularly (most years) C. Occasionally D. Rarely (every few years) E. Never (or not yet)

Ques6on How familiar are you with simula6on methods such as bootstrap confidence intervals and randomiza6on tests? A. Very B. Somewhat C. A liale D. Not at all E. Never heard of them before!

Ques6on

Have you used randomiza6on methods in Intro Stat? A. Yes, as a significant part of the course B. Yes, as a minor part of the course C. No D. What are randomiza6on methods?

Ques6on

Have you used randomiza6on methods in any sta6s6cs class that you teach? A. Yes, as a significant part of the course B. Yes, as a minor part of the course C. No D. What are randomiza6on methods?

Intro Stat – Revise the Topics •  Descrip6ve Sta6s6cs – one and two samples •  Normal distribu6ons •  Data produc6on (samples/experiments)

•  Sampling distribu6ons (mean/propor6on)

•  Confidence intervals (means/propor6ons)

•  Hypothesis tests (means/propor6ons)

•  ANOVA for several means, Inference for regression, Chi-‐square tests

•  Data produc6on (samples/experiments) •  Bootstrap confidence intervals •  Randomiza6on-‐based hypothesis tests •  Normal distribu6ons

•  Bootstrap confidence intervals •  Randomiza6on-‐based hypothesis tests

•  Descrip6ve Sta6s6cs – one and two samples

We need a snack!

What propor-on of Reese’s Pieces are

Orange? Find the propor6on that are orange

for your “sample”.

Propor6on orange in 100 samples of size n=100

BUT – In prac6ce, can we really take lots of samples from the same popula6on?

Bootstrap Distribu-ons

Or: How do we get a sense of a sampling distribu6on when we only have ONE sample?

Suppose we have a random sample of 6 people:

Original Sample

Create a “sampling distribu6on” using this as our simulated popula6on

Bootstrap Sample: Sample with replacement from the original sample, using the same sample size.

Original Sample Bootstrap Sample

Simulated Reese’s Popula6on

Sample from this “popula6on”

Original Sample

Create a bootstrap sample by sampling with replacement from the original sample.

Compute the relevant sta6s6c for the bootstrap sample.

Do this many 6mes!! Gather the bootstrap sta6s6cs all together to form a bootstrap distribu6on.

Original Sample

BootstrapSample

BootstrapSample

BootstrapSample

● ● ●

Bootstrap Sta6s6c

Sample Sta6s6c

Bootstrap Sta6s6c

Bootstrap Sta6s6c

● ● ●

Bootstrap Distribu6on

Example: What is the average price of a used Mustang car?

Select a random sample of n=25 Mustangs from a website (autotrader.com) and record the price (in $1,000’s) for each car.

Sample of Mustangs:

Our best es6mate for the average price of used Mustangs is $15,980, but how accurate is that es6mate?

Price0 5 10 15 20 25 30 35 40 45

MustangPrice Dot Plot

𝑛=25 𝑥 =15.98 𝑠=11.11

Original Sample Bootstrap Sample

We need technology!

Introducing

StatKey. www.lock5stat.com/statkey

StatKey

Std. dev of 𝑥 ’s=2.18

Using the Bootstrap Distribu6on to Get a Confidence Interval – Method #1 The standard devia6on of the bootstrap sta6s6cs es6mates the standard error of the sample sta6s6c.

Quick interval es6mate :

𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ±2∙𝑆𝐸

For the mean Mustang prices: 15.98±2∙2.18=15.98±4.36=(11.62, 20.34)

Using the Bootstrap Distribu6on to Get a Confidence Interval – Method #2

Keep 95% in middle

Chop 2.5% in each tail

Chop 2.5% in each tail

We are 95% sure that the mean price for Mustangs is between $11,930 and $20,238

Bootstrap Confidence Intervals Version 1 (Sta6s6c ± 2 SE): Great prepara6on for moving to tradi6onal methods Version 2 (Percen6les): Great at building understanding of confidence intervals

Playing with StatKey!

See the purple pages in the folder.

Traditional Inference 1. Which formula?

2. Calculate summary stats

5. Plug and chug

𝑥 ± 𝑡↑∗ ∙ 𝑠⁄√𝑛   𝑥 ± 𝑧↑∗ ∙ 𝜎⁄√𝑛  

𝑛=25, 𝑥 =15.98, 𝑠=11.11

3. Find t*

95% CI ⇒ 𝛼⁄2 = 1−0.95/2 =0.025

4. df?

df=25−1=24

OR

t*=2.064

15.98±2.064∙ 11.11⁄√25  

15.98±4.59=(11.39, 20.57)

6. Interpret in context

CI for a mean

7. Check condi6ons

We want to collect some data from you. What should

we ask you for our one quan6ta6ve ques6on and

our one categorical ques6on?

What quan6ta6ve data should we collect from you? A.  What was the class size of the Intro Stat course you

taught most recently? B.  How many years have you been teaching Intro Stat? C.  What was the travel 6me, in hours, for your trip to

Boston for JMM? D.  Including this one, how many 6mes have you aaended

the January JMM? E.  ???

What categorical data should we collect from you? A.  Did you fly or drive to these mee6ngs? B.  Have you aaended any previous JMM mee6ngs? C.  Have you ever aaended a JSM mee6ng? D.  ??? E.  ???

Why does the bootstrap

work?

Sampling Distribu6on

Popula6on

µ

BUT, in prac6ce we don’t see the “tree” or all of the “seeds” – we only have ONE seed

Bootstrap Distribu6on

Bootstrap “Popula6on”

What can we do with just one seed?

Grow a NEW tree!

𝑥 

Es6mate the distribu6on and variability (SE) of 𝑥 ’s from the bootstraps

µ

Golden Rule of Bootstraps

The bootstrap sta3s3cs are to the original sta3s3c

as the original sta3s3c is to the popula3on parameter.

How do we assess student understanding of these methods (even on in-‐class exams without computers)?

See the green pages in the folder.

hap://www.youtube.com/watch?v=3ESGpRUMj9E

Paul the Octopus

hap://www.cnn.com/2010/SPORT/football/07/08/germany.octopus.explainer/index.html

•  Paul the Octopus predicted 8 World Cup games, and predicted them all correctly

•  Is this evidence that Paul actually has psychic powers?

•  How unusual would this be if he were just randomly guessing (with a 50% chance of guessing correctly)?

•  How could we figure this out?

Paul the Octopus

•  Each coin flip = a guess between two teams •  Heads = correct, Tails = incorrect •  Flip a coin 8 6mes and count the number of heads. Remember this number!

Did you get all 8 heads?

(a) Yes (b) No

Simulate!

Let p denote the propor6on of games that Paul guesses correctly (of all games he may have predicted)

H0 : p = 1/2 Ha : p > 1/2

Hypotheses

•  A randomiza3on distribu3on is the distribu6on of sample sta6s6cs we would observe, just by random chance, if the null hypothesis were true •  A randomiza6on distribu6on is created by simula6ng many samples, assuming H0 is true, and calcula6ng the sample sta6s6c each 6me

Randomiza-on Distribu-on

•  Let’s create a randomiza6on distribu6on for Paul the Octopus!

•  On a piece of paper, set up an axis for a dotplot, going from 0 to 8

•  Create a randomiza6on distribu6on using each other’s simulated sta6s6cs

•  For more simula6ons, we use StatKey

Randomiza-on Distribu-on

•  The p-‐value is the probability of geyng a sta6s6c as extreme (or more extreme) as that observed, just by random chance, if the null hypothesis is true

•  This can be calculated directly from the randomiza6on distribu6on!

p-‐value

StatKey

( )81 0.00392 =

•  Create a randomiza6on distribu6on by simula6ng assuming the null hypothesis is true

•  The p-‐value is the propor6on of simulated sta6s6cs as extreme as the original sample sta6s6c

Randomiza-on Test

•  How do we create randomiza6on distribu6ons for other parameters?

•  How do we assess student understanding? •  Connec6ng intervals and tests •  Connec6ng simula6ons to tradi6onal methods

•  Technology for using simula6on methods

•  Experiences in the classroom

Coming Aarac-ons -‐ Friday

Using Randomiza-on Methods to Build Conceptual Understanding

of Sta-s-cal Inference: Day 2

Lock, Lock, Lock, Lock, and Lock MAA Minicourse-‐ Joint Mathema6cs Mee6ngs

San Diego, CA January 2013

Schedule: Day 2 Friday, 1/11, 9:00 – 11:00 am

5. More on Randomiza-on Tests •  How do we generate randomiza6on distribu6ons for various sta6s6cal tests?

•  How do we assess student understanding when using this approach?

6. Connec-ng Intervals and Tests 7. Connec-ng Simula-on Methods to Tradi-onal

8. Technology Op-ons •  Brief soSware demonstra6on (Minitab, R, Excel, more StatKey...)

9. Wrap-‐up •  How has this worked in the classroom? •  Par6cipant comments and ques6ons

10. Evalua-ons

•  In a randomized experiment on treating cocaine addiction, 48 people were randomly assigned to take either Desipramine (a new drug), or Lithium (an existing drug)

•  The outcome variable is whether or not a patient relapsed

•  Is Desipramine signi;icantly better than Lithium at treating cocaine addiction?

Cocaine Addiction

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R

R R R R R R

R R R R R R

R R R R R R

Desipramine Lithium

1. Randomly assign units to treatment groups

R R R R

R R R R R R

R R R R R R

N N N N N N

R R R R R R

R R R R N N

N N N N N N

R R

N N N N N N

R = Relapse N = No Relapse

R R R R

R R R R R R

R R R R R R

N N N N N N

R R R R R R

R R R R R R

R R N N N N

R R

N N N N N N

2. Conduct experiment

3. Observe relapse counts in each group

Lithium Desipramine

10 relapse, 14 no relapse 18 relapse, 6 no relapse

1. Randomly assign units to treatment groups

10 1824

ˆ ˆ

24.333

new oldp p

= −

= −

−

•  Assume the null hypothesis is true •  Simulate new randomizations •  For each, calculate the statistic of interest •  Find the proportion of these simulated statistics that are as extreme as your observed statistic

Randomization Test

R R R R

R R R R R R

R R R R R R

N N N N N N

R R R R R R

R R R R N N

N N N N N N

R R

N N N N N N


R R R R R R

R R R R N N

N N N N N N

N N N N N N

R R R R R R

R R R R R R

R R R R R R

N N N N N N

R N R N

R R R R R R

R N R R R N

R N N N R R

N N N R

N R R N N N

N R N R R N

R N R R R R

Simulate another randomiza6on

Desipramine Lithium


ˆ ˆ16 1224 240.167

N Op p−

= −

=

R R R R

R R R R R R

R R R R R R

N N N N N N

R R R R R R

R N R R N N

R R N R N R

R R

R N R N R R

Simulate another randomiza6on

Desipramine Lithium


ˆ ˆ17 1124 240.250

N Op p−

= −

=

•  Start with 48 cards (Relapse/No relapse) to match the original sample.

• Shuf;le all 48 cards, and rerandomize them into two groups of 24 (new drug and old drug)

•  Count “Relapse” in each group and ;ind the difference in proportions, 𝑝 ↓𝑁 − 𝑝 ↓𝑂 .

•  Repeat (and collect results) to form the randomization distribution.

•  How extreme is the observed statistic of -‐0.33?

Physical Simulation

The observed difference in propor6ons was -‐0.33. How unlikely is this if there is no difference in the drugs? To begin to get a sense of this: How many of you had a randomly simulated difference in propor6ons this extreme?

Was your simulated difference in propor6ons more extreme than the sample sta6s6c (that is, was it less than or equal to -‐0.33)?

A.  Yes B.  No

A randomization sample must: •  Use the data that we have (That’s why we didn’t change any of the results on the cards) AND •  Match the null hypothesis (That’s why we assumed the drug didn’t matter and combined the cards)

Cocaine Addiction

StatKey

The probability of getting results as extreme or more extreme than those observed if the null hypothesis is true, is about .02. p-‐value

Propor6on as extreme as observed sta6s6c

observed sta6s6c

Distribu6on of Sta6s6c Assuming Null is True

How can we do a randomiza-on test for a correla-on?

Is the number of penal6es given to an NFL team posi6vely correlated with the “malevolence” of the team’s uniforms?

Ex: NFL uniform “malevolence” vs. Penalty yards

r = 0.430 n = 28

Is there evidence that the popula6on correla6on is posi6ve?

Key idea: Generate samples that are (a) consistent with the null hypothesis (b) based on the sample data.

H0 : ρ = 0

r = 0.43, n = 28

How can we use the sample data, but ensure that the correla6on is zero?

Randomize one of the variables!

Let’s look at StatKey.

Traditional Inference 1. Which formula?

2. Calculate numbers and plug into formula

3. Plug into calculator

4. Which theore6cal distribu6on?

5. df?

6. find p-‐value

0.01 < p-‐value < 0.02

43.2=

212r

nrt−

−=

243.0122843.0

−

−

How can we do a randomiza-on test for a mean?

Example: Mean Body Temperature

Data: A random sample of n=50 body temperatures.

Is the average body temperature really 98.6oF?

BodyTemp96 97 98 99 100 101

BodyTemp50 Dot Plot

H0:μ=98.6 Ha:μ≠98.6

n = 50 𝑥 =98.26 s = 0.765

Data from Allen Shoemaker, 1996 JSE data set ar6cle

Key idea: Generate samples that are (a) consistent with the null hypothesis (b) based on the sample data.

How to simulate samples of body temperatures to be consistent with H0: μ=98.6?

Randomiza6on Samples How to simulate samples of body temperatures to be consistent with H0: μ=98.6?

1.  Add 0.34 to each temperature in the sample (to get the mean up to 98.6).

2.  Sample (with replacement) from the new data.

3.  Find the mean for each sample (H0 is true). 4.  See how many of the sample means are as

extreme as the observed 𝑥 =98.26.

Let’s try it on

StatKey.

Playing with StatKey!

See the orange pages in the folder.

Choosing a Randomiza6on Method A=Sleep 14 18 11 13 18 17 21 9 16 17 14 15 mean=15.25 B=Caffeine 12 12 14 13 6 18 14 16 10 7 15 10 mean=12.25

Example: Word recall

Op6on 1: Randomly scramble the A and B labels and assign to the 24 word recalls.

H0: μA=μB vs. Ha: μA≠μB

Op6on 2: Combine the 24 values, then sample (with replacement) 12 values for Group A and 12 values for Group B.

Reallocate

Resample

Ques6on

In Intro Stat, how cri6cal is it for the method of randomiza6on to reflect the way data were collected? A. Essen6al B. Rela6vely important C. Desirable, but not impera6ve D. Minimal importance E. Ignore the issue completely

How do we assess student understanding of these methods (even on in-‐class exams without computers)?

See the blue pages in the folder.

Connec6ng CI’s and Tests

Randomiza6on body temp means when μ=98.6

xbar98.2 98.3 98.4 98.5 98.6 98.7 98.8 98.9 99.0

Measures from Sample of BodyTemp50 Dot Plot

97.9 98.0 98.1 98.2 98.3 98.4 98.5 98.6 98.7bootxbar


Bootstrap body temp means from the original sample

What’s the difference?

Fathom Demo: Test & CI

Sample mean is in the “rejec6on region”

⟺ Null mean is outside the confidence interval

What about Tradi6onal Methods?

Transi6oning to Tradi6onal Inference

AFTER students have seen lots of bootstrap distribu6ons and randomiza6on distribu6ons…

Students should be able to •  Find, interpret, and understand a confidence

interval •  Find, interpret, and understand a p-‐value

slope (thousandths)-60 -40 -20 0 20 40 60

Measures from Scrambled RestaurantTips Dot Plot

r-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Measures from Scrambled Collection 1 Dot Plot

Nullxbar98.2 98.3 98.4 98.5 98.6 98.7 98.8 98.9 99.0


Diff-4 -3 -2 -1 0 1 2 3 4

Measures from Scrambled CaffeineTaps Dot Plot

xbar26 27 28 29 30 31 32

Measures from Sample of CommuteAtlanta Dot Plot

Slope :Restaurant 6ps Correla6on: Malevolent uniforms

Mean :Body Temperatures Diff means: Finger taps

Mean : Atlanta commutes

phat0.3 0.4 0.5 0.6 0.7 0.8

Measures from Sample of Collection 1 Dot PlotPropor6on : Owners/dogs

What do you no6ce? All bell-‐shaped distribu6ons!

Bootstrap and Randomiza-on Distribu-ons

The students are primed and ready to learn about the normal distribu-on!

Transi6oning to Tradi6onal Inference

Confidence Interval: 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ± 𝑧↑∗ ∙𝑆𝐸 Hypothesis Test: 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 −𝑁𝑢𝑙𝑙 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑆𝐸

•  Introduce the normal distribu6on (and later t)

•  Introduce “shortcuts” for es6ma6ng SE for propor6ons, means, differences, slope…

z* -‐z*

95%

Confidence Intervals

Test sta6s6c

95%

Hypothesis Tests

Area is p-‐value

Confidence Interval: 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ± 𝑧↑∗ ∙𝑆𝐸 Hypothesis Test: 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 −𝑁𝑢𝑙𝑙 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑆𝐸

Yes! Students see the general paaern and not just individual formulas!

Brief Technology Session Choose One!

R (Kari) Excel (Eric) Minitab (Robin) TI (Pay) More StatKey (Dennis)

(Your binder includes informaFon on using Minitab, R, Excel, Fathom, Matlab, and SAS.)

Student Preferences

Which way of doing inference gave you a better conceptual understanding of con;idence intervals and hypothesis tests?

Bootstrapping and Randomiza-on

Formulas and Theore-cal Distribu-ons

113 51 69% 31%

Student Preferences

Which way did you prefer to learn inference (con;idence intervals and hypothesis tests)?

Bootstrapping and Randomiza-on

Formulas and Theore-cal Distribu-ons

105 60 64% 36%

Simulation Traditional AP Stat 31 36

No AP Stat 74 24

Student Behavior •  Students were given data on the second midterm and asked to compute a con;idence interval for the mean

•  How they created the interval:

Bootstrapping t.test in R Formula

94 9 9 84% 8% 8%

A Student Comment " I took AP Stat in high school and I got a 5. It was mainly all equations, and I had no idea of the theory behind any of what I was doing. Statkey and bootstrapping really made me understand the concepts I was learning, as opposed to just being able to just spit them out on an exam.”

-‐ one of Kari’s students

Thank you for joining us!

More informa-on is available on www.lock5stat.com

Feel free to contact any of us with

any comments or ques-ons.

Please fill out the Minicourse evalua6on form at www.surveymonkey.com/s/JMM2013MinicourseSurvey

using&randomiza-on&methods&to& … · 2014-10-29 ·...

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